Comparison of Two Approaces for Determining Ground-Water Discharge and Pumpage in the Lower Arkansas River Basin Colorado 1997-98

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diffP = log (V)— log (V), (8) where V denotes estimated pumpage as calculated by the PCC approach, and V denotes a corresponding total pumpage as measured by a TFM. As with instan- taneous discharge measurements, a log transformation was used, so that the variable of interest is the differ- ence between the log- transformed values. Because diffP is a random variable like diffQ, the probability distribution must be characterized. However, as mentioned earlier in the report, unlike diffQ, the distribution for diffP deviated significantly from normality. There are a number of data values found outside the range of the majority of data values, and such a deviation from normality can cause serious problems with analysis of variance. Therefore, a rank transformation was performed on the data before performing the analysis, and an inverse rank transfor- mation (linear approximation) to the results of the analysis of variance provided estimates of the central tendency of the distribution of diffP. Use of the rank transformation in analysis of variance is discussed by Iman and Conover (1981), Helsel and Hirsch (1992), Kepner and Wackerly (1996) and Hora and Iman (1988). Rank transformation does not render the test truly nonparametric, but asymptotic normal theory should be more applicable than would be the case if using untransformed data. Rank transformation mini- mizes the influence of very large outliers so that the analysis better reflects the central tendency of the data. Evaluating the data without the influence of extreme outliers was essential in understanding the data, and the results of this analysis indicated the types of errors in estimation of pumpage expected at a typical site under typical circumstances. Because a typical -site analysis is inadequate when analyzing aggregated pumpage for a number of wells, a separate analysis of this problem is discussed later in the report in the section titled "Estimation of Total Network Pumpage" . The overall pattern of differences between V and V are illustrated in figure 9A, which is a plot of diffP versus V, and in figure 9B, which is a plot of the differ- ence in untransformed pumpage versus V. These plots are analogous to the plots in figure 2A and 2B for discharge. Variability about the mean tends to be more nearly constant in figure 9A than in figure 9B for most of the data, so making a logarithmic transformation on the variables is reasonable. These plots also show clearly that there is a small proportion of the differ- ences for which diffP tends to be outside the range of the majority of the data. Boxplots of diffP for all the data pooled and for each level of method, make, and type are shown in figure 10. About 80 percent of the differences in pumpage estimates between the TFM and PCC approach were less than 10 percent, more than 50 percent of the differences were less than 6 percent, and the median difference was about 1 percent (fig. l0A). The distribution of the differences varied somewhat depending on method, make, and type (figs. IOB, IOC, and IOD). The analysis of variance model was first applied using all levels of each of the fixed factors: method, make, and type. Significant differences occurred between all pairs of methods, but not between different makes or types. However, pooling the pumpage data in the same manner as the discharge data allows direct comparisons between results of the two analyses. Such comparisons are useful and can be used to determine how errors in instantaneous discharge measurements affect errors in pumpage calculations. Thus, the analysis was redone using the same pooling described in the "Comparison of Instantaneous Ground -Water Discharge Measurements" section. Diagnostic plots again indicated satisfactory adherence to the analysis of variance assumptions. Final results of the analysis of variance are listed in tables 6 through 8 and are analogous to the results for the instantaneous discharge data presented in tables 1 through 3. As stated earlier, results from the analysis of variance is in terms of ranks, so a linear approximation to the rank - transformation curve near the median was used to back - transform and obtain results in terms of diffP. Differences in estimates of the mean differences listed in tables 6 through 8 that are more than 2 standard errors from zero again are indi- cated as being statistically significant. The overall grand mean difference for all possible pairs of pumpage in table 6 is 0.0001 (0.01 percent), again almost zero. The estimates for the portable flowmeter method effects were: for method C, 0.73 percent; for method M, 0.22 percent; and for method P, —0.93 percent. These effects are comparable in magnitude to portable flowmeter method effects for the well discharge data in table 1. Similarly, signs of the make and type effects are the 26 Comparison of Two Approaches for Determining Ground -Water Discharge and Pumpage in the Lower Arkansas River Basin, Colorado, 1997 -98